An exponential Diophantine equation related to the sum of powers of two consecutive terms of a Lucas sequence and x-coordinates of Pell equations

作     者：Erazo, Harold S. Gomez, Carlos A. 

作者机构：Inst Nacl Matemat Pura & Aplicada Estr Dona Castorina 110 Rio De Janeiro Brazil Univ Valle Dept Matemat Calle 13 100-00 Cali Colombia 

出 版 物：《PERIODICA MATHEMATICA HUNGARICA》 (匈牙利数学学报)

年 卷 期：2021年第83卷第2期

页      面：165-184页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：CAPES (Universidad del Valle) 

主　　题：Pell equation Fibonacci numbers Lower bounds for linear forms in logarithms LLL algorithm 

摘      要：For the Fibonacci sequence the identity F-n(2) + F-n+1(2) = F2n+1 holds for all n = 0. Let X := (X-l)(l = 1) be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation X-2 - dY(2) = +/- 1 corresponding to a nonsquare integer d 1. In this paper, we investigate all positive nonsquare integers d for which there are at least two positive integers X and X of X having a representation as the sum of xth powers of two consecutive terms of a Lucas sequence. Then we solve this problem for Fibonacci numbers.